ABSTRACT
We turn now to the final group of paradoxes. These are much less clearly delineated than those in the other three groups, but also more fundamental. At their root there is a kind of second-order paradox, resting on the backs of all the others so far considered. One radical solution to all of them would be to abandon the concept of the infinite as incoherent. (Without it none of them can properly get off the ground.) So they put collective pressure on us to do that. On the other hand we can feel equally strong pressure from elsewhere to retain the concept. It is true that reflection on the nature of space and time now seems less decisive than it might once have done, because, now that we have greater scientific insight, we are no longer sure that either space or time is infinitely big (infinite by addition) or infinitely divisible (infinite by division).10 Still, it at least seems to make perfectly good sense, mathematically, to suppose that they are, even if it is false; and this is enough for the concept of the infinite to be coherent. Again, consider the natural numbers: whether or not they can be collected together into a single set, we surely want to be able to say that there are infinitely many of them. But perhaps the strongest pressure to retain the concept of the infinite comes from a rather nebulous, though powerful, sense of our own finitude. This is something which cuts deeper than our awareness that we are mortal and limited in size, constrained in various ways, and ignorant of so much (though it incorporates all of these). It is a sense of being cast into a world that is completely independent of us, most of which confronts us as something alien, something other than us, something that impinges on us from without and limits us. (I am not denying that there can be value in overcoming this sense. I shall return to this point at the very end of the book.) This instils in us the idea of a contrast: the idea that the world as a whole-the universe-cannot, in its self-contained totality, be similarly limited by something beyond it, because it includes everything. It must be infinite. One of the paradoxes of thought about the infinite, then, is that there are reasons both for and against admitting the concept of infinity.
